Two similar magnets of magnetic moments `M_(1)` and `M_(2)` are taken and vibrated in vibration magnetometer with their like pole together and unlike pole together . If ratio of `(M_(1))/(M_(2)) = (5)/(3)` , then ratio of the period is

Options

A. `(T_(1))/(T_(2)) = (1)/(2)`
B. `(T_(1))/(T_(2)) = 2`
C. `(T_(1))/(T_(2)) = 1`
D. `(T_(1))/(T_(2)) = (1)/(4)`

Answer

To solve the problem, we need to find the ratio of the periods of two similar magnets with given magnetic moments when they are vibrated in a vibration magnetometer with their like poles together and unlike poles together.

### Step-by-Step Solution:

1. **Understanding the Magnetic Moments**:
Given that the ratio of the magnetic moments is:
\[ \frac{M_1}{M_2} = \frac{5}{3} \]

2. **Time Period of a Magnet**:
The time period \( T \) of a magnet in a magnetic field is given by the formula:
\[ T \propto \sqrt{\frac{I}{MB}} \]
where \( I \) is the moment of inertia, \( M \) is the magnetic moment, and \( B \) is the magnetic field. For our purposes, we can say that the time period is inversely proportional to the square root of the magnetic moment.

3. **Case 1: Like Poles Together**:
When the like poles (North-North or South-South) are together, the resultant magnetic moment \( M_r \) is:
\[ M_r = M_1 + M_2 \]
The time period \( T_1 \) is then:
\[ T_1 \propto \frac{1}{\sqrt{M_1 + M_2}} \]

4. **Case 2: Unlike Poles Together**:
When the unlike poles (North-South) are together, the resultant magnetic moment \( M_r \) is:
\[ M_r = |M_1 - M_2| \]
The time period \( T_2 \) is then:
\[ T_2 \propto \frac{1}{\sqrt{|M_1 - M_2|}} \]

5. **Calculating the Resultant Magnetic Moments**:
Substitute \( M_1 = 5k \) and \( M_2 = 3k \) (where \( k \) is a constant):
- For like poles:
\[ M_r = 5k + 3k = 8k \]
- For unlike poles:
\[ M_r = |5k - 3k| = 2k \]

6. **Finding the Ratio of Time Periods**:
Now, we can find the ratio of the time periods:
\[ \frac{T_1}{T_2} = \frac{\sqrt{|M_1 - M_2|}}{\sqrt{M_1 + M_2}} = \frac{\sqrt{2k}}{\sqrt{8k}} = \frac{\sqrt{2}}{\sqrt{8}} = \frac{\sqrt{2}}{2\sqrt{2}} = \frac{1}{2} \]

7. **Final Result**:
Therefore, the ratio of the periods is:
\[ \frac{T_1}{T_2} = \frac{1}{2} \]

### Conclusion:
The ratio of the periods when the magnets are vibrated with their like poles together and unlike poles together is \( \frac{1}{2} \).